Mechanical resonators are physical structures that are designed to vibrate at high frequencies. Such resonators may be incorporated into a variety of devices such as timing oscillators, mass sensors, gyros, accelerometers, switches, and electromagnetic fuel sensors, amongst others.
During use, mechanical resonators, and the devices which incorporate the same, may be exposed to different temperature conditions and variations. Such conditions and variations can cause material expansion and contraction, as well as changes in material stiffness. This can result in a variation in vibrational characteristics (e.g., resonating frequency) across the temperature range. These effects also can lead to increased noise, reduction in bandwidth, deterioration of signal quality and can, in general, create stability problems in devices.
The temperature stability of a mechanical resonator may be quantified as the temperature coefficient of frequency (TCF), which is expressed as: TCF=(1/f)(∂f/∂T), where f is the resonance frequency and T is the temperature. Another term that is used to quantify the stiffness component of the temperature stability of a mechanical resonator (which is one of the primary contributors to TCF) is the temperature coefficient of stiffness (TCS), which can be expressed as: TCS=(1/Ceff)(∂Ceff/∂T), where Ceff is the effective stiffness coefficient of the resonator.
To address the effects resulting from temperature change, it can be advantageous for mechanical resonating structures to have temperature compensation capabilities to improve the stability of such structures, and associated devices, over a range of temperatures.